Theorem raleqbidvv 3334

Description: Version of raleqbidv 3346 with additional disjoint variable conditions, not requiring ax-8 2110 nor df-clel 2816. (Contributed by BJ, 22-Sep-2024.)

Hypotheses

Ref	Expression
raleqbidvv.1	⊢ (𝜑 → 𝐴 = 𝐵)
raleqbidvv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
raleqbidvv	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidvv

Step	Hyp	Ref	Expression
1		raleqbidvv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbidvv.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	raleqbidva 3332	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ral 3062  df-rex 3071

This theorem is referenced by:  rexeqbidvvOLD  3337  raleqbi1dv  3338  gpg5nbgrvtx03star  48036  postcposALT  49172